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“PRIMES Is In P”

“PRIMES Is In P”. Jeremy Thomas. “The AKS Algorithm”. (An Explanation of the Title). “PRIMES”. The decisional problem of determining whether or not a given integer n is prime. “P”.

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“PRIMES Is In P”

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  1. “PRIMES Is In P” Jeremy Thomas “The AKS Algorithm”

  2. (An Explanation of the Title) “PRIMES” • The decisional problem of determining whether or not a given integer n is prime.

  3. “P” • The class of decisional problems for which any problem can be solved by a deterministic algorithm in time that can be bounded by a polynomial on the size of the input of the problem.

  4. AKS Algorithm Given n. We can find if n is prime or not in time bounded by a polynomial The size in bits of the input number n. This is unconditional!! (Will always return the right answer)

  5. The General Approach Example

  6. To make this evaluation more feasible, take the equation modulo a polynomial Therefore if n is prime, this holds for all values of r and a.

  7. From here we chose a reasonable r: • A value of r is reasonable if • r is a prime number • r-1 has a large prime factor • q | k where Now, we evaluate the congruence for a “small” number of a’s

  8. The Algorithm Input: Integer n > 1 if (n has the form a^b with b > 1) then output COMPOSITE r := 2 while (r < n) { if (gcd(n,r) is not 1) then output COMPOSITE if (r is prime greater than 2) then { let q be the largest factor of r-1 if (q > 4sqrt(r)log n) and (n^((r-1)/q) is not 1 (mod r)) then break } r := r+1 } for a = 1 to 2sqrt(r)log n { if ( (x-a)^n is not (x^n-a) (mod x^r-1,n) ) then output COMPOSITE } output PRIME;

  9. Time Complexity Analysis “Soft oh”

  10. Step 1: • -straight reference to Gathen ??. • Step 2: • check gcd (r,n) • makes a recursive call to check r prime • Find large prime factor • Step 3: • - sieve for all “needed” value’s of a Roughly a polynomial of order twelve in the size of the input.

  11. The Proof ?

  12. The Implementation Lets look at code!!

  13. Conclusion • Monumental theoretical accomplishment. • - Something for math folks to talk about • - Catalyst for further research. • No practical relevance. • - The run time for the algorithm is slow. • - Unlisted constants in the big O evaluation are the difference.

  14. References • “Primes Is In P”. Manindra Agrawal, Neeraj Kayal and Nitin Saxena. (http://www.cse.iitk.ac.in/news/primality.html) • Joachim von zur Gathen and Jürgen Gerhard. Modern Computer Algebra. Cambridge University Press, 1999. • Prime Pages. (http://www.utm.edu/research/primes/prove/prove4_3.html) • The PRIMES is in P little FAQ. • (http://crypto.cs.mcgill.ca/~stiglic/PRIMES_P_FAQ.html#references) • Prime is in P.(http://xocxoc.home.att.net/math/primes.htm)

  15. Questions?

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